How is the period of a simple pendulum growing with increasing amplitude?
Lampret, Vito
For the period ▫$T(\alpha)$▫ of a simple pendulum with the length ▫$L$▫ and the amplitude (the initial elongation) ▫$\alpha\in (0,\pi)$▫, a strictly increasing sequence ▫$T_n(\alpha)$▫ is constructed ...such that the relations ▫$$\begin{aligned} & T_1(\alpha) = 2\sqrt{\frac{L}{g}} \left[ \pi-2+\frac{1}{\epsilon} \ln \left( \frac{1+\epsilon}{1-\epsilon} \right) + \left( \frac{\pi}{4}-\frac{2}{3} \right) \epsilon^2 \right],\\ & T_{n+1}(\alpha) = T_n(\alpha) + 2\sqrt{\frac{L}{g}} \left( \pi w_{n+1}^2 - \frac{2}{2n+3} \right) \epsilon^{2n+2}, \end{aligned}$$▫ and ▫$$ 0 < \frac{T(\alpha) - T_n(\alpha)}{T(\alpha)} < \frac{2\epsilon^{2n+2}}{\pi(2n+1)}, $$▫ holds true, for ▫$\alpha\in (0,\pi)$▫, ▫$n\in\mathbb{N}$▫, ▫$w_n:=\prod_{k=1}^n \frac{2k-1}{2k}$▫ (the ▫$n$▫th Wallis' ratio) and ▫$\varepsilon=\sin(\alpha/2)$▫.
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